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Some Methods of Infinite Dimensional Analysis in Hydrodynamics: An Introduction

  • Sergio Albeverio
  • Benedetta Ferrario
Part of the Lecture Notes in Mathematics book series (LNM, volume 1942)

In these lectures we shall concentrate on certain mathematical results concerning the case of deterministic Euler and stochastic Navier-Stokes equations for incompressible fluids. For general references we refer to [Fri95], [Tem83], [CF88], [VF88], [VKF79], [Lio96], [MP94], [NPe01], [Che96b], [Che98], [Che04], [CK04a], [Con95], [Con94], [Con01a], [Con01b], [FMRT01], [LR02], [MB02] [Tem84] and [Bir60] and for a discussion of challenging open problems see, e.g. [Fef06], [Can00], [Can04], [CF03], [Cho94], [Con95], [Con01a], [FMRT01], [Gal01], [ES00a], [Hey90], [Ros06] and [FMB03]. We shall concentrate particularly on the study of invariant measures associated with the above equations for fluids. On the one hand, this follows an analogy with the statistical mechanical approach to classical particle systems and ergodic theory, see, e.g. [Min00], [Rue69]. On the other hand, it follows Kolmogorov's suggestion, see e.g. [ER85], of adding small stochastic perturbations (“noise”) in classical dynamical systems, so to construct invariant measures and then study what happens when removing the noise.

The content of our lecture is as follows: in Section 2 we shall study the deterministic Euler equation and construct certain natural invariant measures for it. We also relate this analysis with the study of a certain Hamiltonian system describing vortices (“vortex models”). In Section 3 we shall study the stochastic Navier-Stokes equation with Gaussian space-time white noise and its invariant measure. We also provide brief comments and bibliographical references concerning recent work in directions which are complementary to those described here.

Keywords

Stokes Equation Invariant Measure Euler Equation Dimensional Analysis Besov Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. AB02.
    S. Albeverio and Ya. Belopolskaya. Probabilistic approach to hydrodynamic equations. In Probabilistic methods in fluids, pages 1–21. World Sci. Publ., River Edge, NJ, Swansea, UK, April 2002.Google Scholar
  2. AB06.
    S. Albeverio and Ya. Belopolskaya. Probabilistic approach to systems of nonlinear PDEs and vanishing viscosity method. Markov Process. Relat. Fields, 12(1):59–94, 2006.zbMATHMathSciNetGoogle Scholar
  3. ABF.
    S. Albeverio, V. Barbu, and B. Ferrario. Uniqueness of the generators of the 2D Euler and Navier-Stokes flows. Stochastic Processes Appl., in press, available online 27 December 2007.Google Scholar
  4. ABGS00.
    K. Aoki, C. Bardos, F. Golse, and Y. Sone. Derivation of hydrodynamic limits from either the Liouville equation or kinetic models: study of an example. Sūrikaisekikenkyūsho Kōkyūroku, (1146):154–181, 2000. Mathematical analysis of liquids and gases (Japanese) (Kyoto, 1999).Google Scholar
  5. ABHK85.
    S. Albeverio, Ph. Blanchard, and R. Høegh-Krohn. Reduction of nonlinear problems to Schrödinger or heat equations: formation of Kepler orbits, singular solutions for hydrodynamical equations. In S. Albeverio et al., eds, Stochastic aspects of classical and quantum systems (Marseille, 1983), volume 1109 of Lecture Notes in Math., pages 189–206. Springer, Berlin, 1985.CrossRefGoogle Scholar
  6. ABR.
    S. Albeverio, V. Barbu, and M. Röckner. in preparation.Google Scholar
  7. ABW.
    S. Albeverio, Z. Brzeźniak, and J. L. Wu. Stochastic Navier–Stokes equations driven by non Gaussian white noise. in preparation.Google Scholar
  8. AC90.
    S. Albeverio and A. B. Cruzeiro. Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two-dimensional fluids. Comm. Math. Phys., 129(3):431–444, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  9. AF02a.
    S. Albeverio and B. Ferrario. Invariant measures of Lévy-Khinchine type for 2D fluids. In I. M. Davies, N. Jacob, A. Truman, O. Hassan, K. Morgan, and N. P. Weatherill, editors, Probabilistic methods in fluids, pages 130–143, Swansea, UK, April 2002. University of Wales, World Sci. Publ., River Edge, NJ.Google Scholar
  10. AF02b.
    S. Albeverio and B. Ferrario. Uniqueness results for the generators of the two-dimensional Euler and Navier–Stokes flows. The case of Gaussian invariant measures. J. Funct. Anal., 193(1):77–93, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  11. AF03.
    S. Albeverio and B. Ferrario. 2D vortex motion of an incompressible ideal fluid: the Koopman-von Neumann approach. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6(2):155–165, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  12. AF04a.
    S. Albeverio and B. Ferrario. Invariant Gibbs measures for the 2D vortex motion of fluids. S. Albeverio et al. (eds.), Recent developments in stochastic analysis and related topics. Proceedings of the first Sino-German conference on stochastic analysis (a satellite conference of ICM 2002), Beijing, China, 29 August – 3 September 2002. River Edge, NJ: World Scientific. 31–44, 2004.Google Scholar
  13. AF04b.
    S. Albeverio and B. Ferrario. Uniqueness of solutions of the stochastic Navier–Stokes equation with invariant measure given by the enstrophy. Ann. Probab., 32(2):1632–1649, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  14. AFY04.
    S. Albeverio, B. Ferrario, and M. W. Yoshida. On the essential self-adjointness of Wick powers of relativistic fields and of fields unitary equivalent to random fields. Acta Appl. Math., 80(3):309–334, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  15. AGY05.
    S. Albeverio, H. Gottschalk and M. W. Yoshida. Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory. Rev. Math. Phys. 17(2):175–226, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  16. AHK73.
    S. Albeverio and R. Høegh-Krohn. Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions. Comm. Math. Phys., 30:171–200, 1973.MathSciNetCrossRefGoogle Scholar
  17. AHK89.
    S. Albeverio and R. Høegh-Krohn. Stochastic flows with stationary distribution for two-dimensional inviscid fluids. Stochastic Process. Appl., 31(1):1–31, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  18. AHKFL86.
    S. Albeverio, R. Høegh-Krohn, J. E. Fenstad and T. Lindstrøm. Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986.zbMATHGoogle Scholar
  19. AHKM85.
    S. Albeverio, R. Høegh-Krohn, and D. Merlini. Euler flows, associated generalized random fields and Coulomb systems. In Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983), volume 124 of Res. Notes in Math., pages 216–244. Pitman, Boston, MA, 1985.Google Scholar
  20. AK98.
    V. I. Arnol′d and B. A. Khesin. Topological methods in hydrodynamics. Springer-Verlag, New York, 1998.Google Scholar
  21. AKR92.
    S. Albeverio, Yu. G. Kondratiev, and M. Röckner. An approximate criterium of essential selfadjointness of Dirichlet operators. Potential Anal., 1(3):307–317, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  22. AKR95.
    S. Albeverio, Yu. G. Kondratiev, and M. Röckner. Dirichlet operators via stochastic analysis. J. Funct. Anal., 128(1):102–138, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  23. AKR98a.
    S. Albeverio, Yu. G. Kondratiev, and M. Röckner. Analysis and geometry on configuration spaces. J. Funct. Anal., 154(2):444–500, 1998.CrossRefGoogle Scholar
  24. AKR98b.
    S. Albeverio, Yu. G. Kondratiev, and M. Röckner. Analysis and geometry on configuration spaces: the Gibbsian case. J. Funct. Anal., 157(1):242–291, 1998.CrossRefGoogle Scholar
  25. Alb00.
    S. Albeverio. Introduction to the theory of Dirichlet forms and applications. in S. Albeverio, W. Schachermeyer, M. Talagrand, St Flour Lectures on Probability and Statistics. LN Math. 1816, Springer, Berlin (2003), 2000.Google Scholar
  26. ALZ06.
    S. Albeverio, S. Liang, and B. Zegarlinski. Remark on the integration by parts formula for the φ3 4-quantum field model. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9(1):149–154, 2006.zbMATHMathSciNetCrossRefGoogle Scholar
  27. AMS94.
    S. Albeverio, S. A. Molchanov, and D. Surgailis. Stratified structure of the Universe and Burgers’ equation—a probabilistic approach. Probab. Theory Related Fields, 100(4):457–484, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  28. AR95.
    S. Albeverio and M. Röckner. Dirichlet form methods for uniqueness of martingale problems and applications. In Stochastic analysis (Ithaca, NY, 1993), volume 57 of Proc. Sympos. Pure Math., pages 513–528. Amer. Math. Soc., Providence, RI, 1995.Google Scholar
  29. AR96.
    S. Albeverio and F. Russo. Stochastic partial differential equations, infinite dimensional stochastic processes and random fields: A short introduction. L. Vazquez et al. (eds.), Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrödinger systems: theory and applications, Madrid, Spain, September 25–30, 1995. Singapore: World Scientific. 68-86, 1996.Google Scholar
  30. ARdFHK79.
    S. Albeverio, M. Ribeiro de Faria, and R. Høegh-Krohn. Stationary measures for the periodic Euler flow in two dimensions. J. Statist. Phys., 20(6):585–595, 1979.MathSciNetCrossRefGoogle Scholar
  31. ASZ82.
    V. I. Arnold, S. F. Shandarin, and Ya. B. Zeldovich. The large scale structure of the Universe. I. General properties. One- and two-dimensional models. Geophys. Astrophys. Fluid Dyn., 20:111–130, 1982.zbMATHCrossRefGoogle Scholar
  32. BA94.
    M. Ben-Artzi. Global solutions of two-dimensional Navier–Stokes and Euler equations. Arch. Rational Mech. Anal., 128(4):329–358, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  33. Bar98.
    V. Barbu. Optimal control of Navier–Stokes equations with periodic inputs. Nonlinear Anal., Theory Methods Appl., 31(1–2):15–31, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  34. BCF92.
    Z. Brzeźniak, M. Capiński, and F. Flandoli. Stochastic Navier–Stokes equations with multiplicative noise. Stochastic Anal. Appl., 10(5):523–532, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  35. BCJL94.
    L. Bertini, N. Cancrini, and G. Jona-Lasinio. The stochastic Burgers equation. Comm.Math.Phys., 165(2):211, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  36. BCM88.
    V.S. Borkar, R.T. Chari, and S.K. Mitter. Stochastic quantization of field theory in finite and infinite volume. J. Funct. Anal., 81(1): 184–206, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  37. BDPD04.
    V. Barbu, G. Da Prato, and A. Debussche. Essential m-dissipativity of Kolmogorov operators corresponding to periodic 2D-Navier–Stokes equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15(1):29–38, 2004.zbMATHMathSciNetGoogle Scholar
  38. BDS04.
    Yu. Yu. Bakhtin, E. I. Dinaburg, and Ya. Sinai. On solutions with infinite energy and enstrophy of the Navier–Stokes system. Uspekhi Mat. Nauk, 59(6(360)):55–72, 2004.Google Scholar
  39. Bes99.
    H. Bessaih. Martingale solutions for stochastic Euler equations. Stochastic Anal. Appl., 17(5):713–725, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  40. BF80.
    C. Boldrighini and S. Frigio. Equilibrium states for a plane incompressible perfect fluid. Comm. Math. Phys., 72(1):55–76, 1980.zbMATHMathSciNetCrossRefGoogle Scholar
  41. BF99.
    H. Bessaih and F. Flandoli. 2-D Euler equation perturbed by noise. Nonlinear Differential Equations Appl., 6(1):35–54, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  42. BF00.
    H. Bessaih and F. Flandoli. Weak Attractor for a Dissipative Euler Equation. Journal of Dynamics and Differential Equations, 12(4):713–732, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  43. BF81.
    C. Boldrighini and S. Frigio. Erratum: “Equilibrium states for a plane incompressible perfect fluid”. Comm. Math. Phys., 78(2):303, 1980/81.MathSciNetCrossRefGoogle Scholar
  44. BFR05.
    B. Busnello, F. Flandoli, and M. Romito. A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations. Proc. Edinb. Math. Soc., II. Ser., 48(2):295–336, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  45. BG96.
    H. A. Biagioni and T. Gramchev. On the 2D Navier–Stokes equation with singular initial data and forcing term. Mat. Contemp., 10:1–20, 1996.zbMATHMathSciNetGoogle Scholar
  46. Bir60.
    G. Birkhoff. Hydrodynamics: A study in logic, fact and similitude. Revised ed. Princeton Univ. Press, Princeton, N.J., 1960.zbMATHGoogle Scholar
  47. BKL01.
    J. Bricmont, A. Kupiainen, and R. Lefevere. Ergodicity of the 2D Navier–Stokes equations with random forcing. Comm. Math. Phys., 224(1):65–81, 2001. Dedicated to Joel L. Lebowitz.zbMATHMathSciNetCrossRefGoogle Scholar
  48. BL76.
    J. Bergh and J. Löfström. Interpolation spaces. An introduction. Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.Google Scholar
  49. BL04.
    Z. Brzeźniak and Y. Li. Asymptotic behaviour of solutions to the 2D stochastic Navier–Stokes equations in unbounded domains – new developments. Albeverio, S. et al. eds, Recent developments in stochastic analysis and related topics. Proceedings of the first Sino-German conference on stochastic analysis (a satellite conference of ICM 2002), Beijing, China, 29 August – 3 September 2002. River Edge, NJ: World Scientific. 78–111, 2004.Google Scholar
  50. BLS05.
    V. Betz, J. Lörinczi, and H. Spohn. Gibbs measures on Brownian paths: theory and applications. J.-D. Deuschel et al. (eds.), Interacting stochastic systems. Berlin: Springer. 75–102, 2005.CrossRefGoogle Scholar
  51. BP01.
    Z. Brzeźniak and S. Peszat. Stochastic two dimensional Euler equations. Ann. Probab., 29(4):1796–1832, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  52. BR01.
    V. I. Bogachev and M. Röckner. Elliptic equations for measures on infinite dimensional spaces and applications. Probab. Theory Relat. Fields, 120(4):445–496, 2001.zbMATHCrossRefGoogle Scholar
  53. Bre99.
    Y. Brenier. Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math., 52(4):411–452, 1999.MathSciNetCrossRefGoogle Scholar
  54. BRV01.
    S. Benachour, B. Roynette and P. Vallois. Branching process associated with 2d-Navier Stokes equation. Rev. Mat. Iberoamericana 17(2):331–373, 2001.zbMATHMathSciNetGoogle Scholar
  55. BT73.
    A. Bensoussan and R. Temam. Équations stochastiques du type Navier–Stokes. J. Functional Analysis, 13:195–222, 1973.zbMATHMathSciNetCrossRefGoogle Scholar
  56. Bus99.
    B. Busnello. A probabilistic approach to the two-dimensional Navier–Stokes equations. Ann. Probab., 27(4):1750–1780, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  57. Caf89.
    R. E. Caflisch. Mathematical analysis of vortex dynamics. Mathematical aspects of vortex dynamics (Leesburg, VA, 1988), 1–24, SIAM, Philadelphia, PA, 1989.Google Scholar
  58. Can00.
    M. Cannone. Advances in mathematical fluid mechanics (J. Malek, J. Necas, M. Rokyta): Viscous flows in Besov spaces. pages 1–34, 2000.Google Scholar
  59. Can04.
    M. Cannone. Harmonic analysis tools for solving the incompressible Navier–Stokes equations. Handbook of mathematical fluid dynamics, 3:161–244, 2004.MathSciNetCrossRefGoogle Scholar
  60. Car03.
    T. Caraballo. The long-time behaviour of stochastic 2D-Navier–Stokes equations. Davies, I. M. (ed.) et al., Probabilistic methods in fluids. Proceedings of the Swansea 2002 workshop, Wales, UK, April 14–19, 2002. Singapore: World Scientific. 70–83, 2003.Google Scholar
  61. CC95.
    M. Capiński and N. J. Cutland. Nonstandard methods for stochastic fluid mechanics, volume 27 of Series on Advances in Mathematics for Applied Sciences. World Scientific Publishing Co. Inc., River Edge, NJ, 1995.zbMATHGoogle Scholar
  62. CC99.
    M. Capiński and N. J. Cutland. Stochastic Euler equations on the torus. Ann. Appl. Probab., 9(3):688–705, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  63. CC06.
    N. V. Chemetov and F. Cipriano. The 2D Euler equations and the statistical transport equations. Commun. Math. Phys., 267:543–558, 2006.zbMATHMathSciNetCrossRefGoogle Scholar
  64. CDG85.
    S. Caprino and S. De Gregorio. On the statistical solutions of the two-dimensional, periodic Euler equation. Math. Methods Appl. Sci., 7(1):55–73, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  65. CE06.
    N. J. Cutland and B. Enright. Stochastic nonhomogeneous incomressible Navier–Stokes equations. J. Diff. Eq., 228:140–170, 2006.zbMATHMathSciNetCrossRefGoogle Scholar
  66. CF88.
    P. Constantin and C. Foias. Navier–Stokes equations. Chicago Lectures in Mathematics. Chicago, IL etc.: University of Chicago Press. ix, 190 p., 1988.Google Scholar
  67. CF03.
    M. Cannone and S. Friedlander. Navier: blow up and collapse. Notices AMS, 50(1):7–13, 2003.zbMATHMathSciNetGoogle Scholar
  68. CFM07.
    A.-B. Cruzeiro, F. Flandoli, and P. Malliavin. Brownian motion on volume preserving diffeomorphisms group and existence of global solutions of 2D stochastic Euler equation. J. Funct. Anal., 242(1):304–326, 2007.zbMATHMathSciNetCrossRefGoogle Scholar
  69. CFMT83.
    P. Constantin, C. Foiaş, O. Manley, and R. Temam. Connexion entre la théorie mathématique des équations de Navier–Stokes et la théorie conventionnelle de la turbulence. C.R.Acad.Sci.Paris Ser:I.Math, 297(11):599–602, 1983.zbMATHMathSciNetGoogle Scholar
  70. CG94.
    M. Capiński and D. Gatarek. Stochastic equations in Hilbert space with application to Navier–Stokes equations in any dimension. J. Funct. Anal., 126(1):26–35, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  71. CH97.
    A. B. Cruzeiro and Z. Haba. Invariant measure for a wave equation on a Riemannian manifold. Stochastic differential and difference equations (Györ, 1996), 35–41, Progr. Systems Control Theory, 23, Birkhäuser Boston, Boston, MA, 1997.Google Scholar
  72. Cha96.
    M. H. Chang. Large deviation for Navier–Stokes equation with small stochastic perturbation. Applied Mathematics and Computation, 1996.Google Scholar
  73. Che96a.
    J.-Y. Chemin. A remark on the inviscid limit for two-dimensional incompressible fluids. Commun. Partial Differ. Equations, 21(11–12): 1771–1779, 1996.zbMATHMathSciNetGoogle Scholar
  74. Che96b.
    J.-Y. Chemin. About Navier–Stokes system. Publication du Laboratoire d’Analyse Numérique R 96023, 1996.Google Scholar
  75. Che98.
    J.-Y. Chemin. Perfect incompressible fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1998. Translated from the 1995 French original by I. Gallagher and D. Iftimie.Google Scholar
  76. Che04.
    J.-Y. Chemin. The incompressible Navier–Stokes system seventy years after Jean Leray. (Le système de Navier–Stokes incompressible soixante dix ans après Jean Leray.). Guillopé, L. (ed.) et al., Proceedings of the colloquium dedicated to the memory of Jean Leray, Nantes, France, June 17–18, 2002. Paris: Société Mathématique de France. Séminaires et Congrès 9, 99–123, 2004.Google Scholar
  77. Cho78.
    P. L. Chow. Stochastic partial differential equations in turbulence related problems. In Probabilistic analysis and related topics, Vol. 1, pages 1–43. Academic Press, New York, 1978.Google Scholar
  78. Cho94.
    A. J. Chorin. Vorticity and turbulence. Applied Mathematical Sciences, 103, 1994.Google Scholar
  79. Cip99.
    F. Cipriano. The two-dimensional Euler equation: a statistical study. Comm. Math. Phys., 201(1):139–154, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  80. CK04a.
    M. Cannone and G. Karch. Smooth or singular solutions to the Navier–Stokes system? J. Differ. Equations, 197(2):247–274, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  81. CK04b.
    N. J. Cutland and H. J. Keisler. Global attractors for 3-dimensional stochastic Navier–Stokes equations. J. Dyn. Differ. Equations, 16(1): 205–266, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  82. CLMP92.
    E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti. A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. Comm. Math. Phys., (143):501–525, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  83. CMR98.
    T. Clopeau, A. Mikelić, and R. Robert. On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity, 11(6):1625–1636, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  84. Con94.
    P. Constantin. Geometric statistic in turbulence. SIAM Rev. 36(1):73–98, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  85. Con95.
    P. Constantin. A few results and open problems regarding incompressible fluids. Notices AMS, 42:658–663, 1995.zbMATHMathSciNetGoogle Scholar
  86. Con01a.
    P. Constantin. Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics, volume 47 of Mathematics unlimited. Springer, Berlin, 2001.Google Scholar
  87. Con01b.
    P. Constantin. Three lectures on mathematical fluid mechanics. Robinson, James C. (ed.) et al., From finite to infinite dimensional dynamical systems. Proceedings of the NATO advanced study institute, Cambridge, UK, August 21-September 1, 1995. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 19, 145–175, 2001.Google Scholar
  88. Cru89a.
    A. B. Cruzeiro. Invariant measures for Euler and Navier–Stokes systems. In stochastic analysis, path integration and dynamics (Warwick, 1987). Pitman Res. Notes Math. Ser., 200:73–82, 1989.MathSciNetGoogle Scholar
  89. Cru89b.
    A. B. Cruzeiro. Solutions et mesures invariantes pour des équations du type Navier–Stokes. Expo.Math (7):73–82, 1989.zbMATHMathSciNetGoogle Scholar
  90. Cut03.
    N. J. Cutland. Stochastic Navier–Stokes equations: Loeb space techniques and attractors. Davies, I. M. (ed.) et al., Probabilistic methods in fluids. Proceedings of the Swansea 2002 workshop, Wales, UK, April 14–19, 2002. Singapore: World Scientific. 97–114, 2003.Google Scholar
  91. Deb02.
    A. Debussche. The 2D-Navier–Stokes equations perturbed by a delta correlated noise. In I. M. Davies, N. Jacob, A. Truman, O. Hassan, K. Morgan, and N. P. Weatherill, editors, Probabilistic methods in fluids, pages 115–129, Swansea, UK, April 2002. University of Wales, World Sci. Publ., River Edge, NJ.Google Scholar
  92. DG95.
    Charles R. Doering and J. D. Gibbon. Applied analysis of the Navier–Stokes equations. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge Univ. Press. xiii, 217 p., 1995.Google Scholar
  93. DP82.
    D. Dürr and M. Pulvirenti. On the vortex flow in bounded domains. Comm. Math. Phys., 85(2):265–273, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  94. DP04.
    G. Da Prato. Kolmogorov equations for stochastic PDEs. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2004.zbMATHGoogle Scholar
  95. DPD99.
    G. Da Prato and A. Debussche. Maximal dissipativity of the Dirichlet operator to the Burgers equation. CMS Conf. Proc., 28, Amer. Math. Soc., pages 85–98, 1999.Google Scholar
  96. DPD02.
    G. Da Prato and A. Debussche. Two-dimensional Navier–Stokes equations driven by a space-time white noise. J. Funct. Anal., 196(1): 180–210, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  97. DPD03.
    G. Da Prato and A. Debussche. Ergodicity for the 3D stochastic Navier–Stokes equations. J.Math.Pures.Appl., 82(8):877–947, 2003.zbMATHMathSciNetGoogle Scholar
  98. DPD07.
    G. Da Prato and A. Debussche. m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Anal., 26(1):31–55, 2007.zbMATHMathSciNetCrossRefGoogle Scholar
  99. DPDT05.
    G. Da Prato, A. Debussche and L. Tubaro. Coupling for some partial differential equations driven by white noise. Stoch. Proc. Appl. 115(8): 1384–1407, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  100. DPDT94.
    G. Da Prato, A. Debussche, and R. Temam. Stochastic Burgers’ equation. NoDEA, Nonlinear Differ. Equ. Appl., 1(4):389–402, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  101. DPT00.
    G. Da Prato and L. Tubaro. Selfadjointness of some infinite-dimensional elliptic operators and application to stochastic quantization. Probab. Theory Relat. Fields, 118(1):131–145, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  102. DPT05.
    G. Da Prato and L. Tubaro. An introduction to the 2D renormalization. preprint, 2005.Google Scholar
  103. DPZ92.
    G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.zbMATHGoogle Scholar
  104. DV87.
    G. Dore and A. Venni. On the closedness of the sum of two closed operators. Math. Z., 196(2):189–201, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  105. E01.
    W. E. Selected problems in materials science. Engquist, Björn (ed.) et al., Mathematics unlimited – 2001 and beyond. Berlin: Springer. 407–432, 2001.Google Scholar
  106. Ebe99.
    A. Eberle. Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, volume 1718 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999.zbMATHGoogle Scholar
  107. Ebi84.
    D. G. Ebin. A concise presentation of the Euler equations of hydrodynamics. Comm. Partial Differential Equations, 9(6):539–559, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  108. EKMS00.
    W. E. Weinan, K. Khanin, A. Mazel, and Ya. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math., 151(3):877–960, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  109. EMS01.
    W. E. Weinan, J. C. Mattingly, and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Comm. Math. Phys., 224(1):83–106, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  110. ER85.
    J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev. Modern Phys., 57(3, part 1):617–656, 1985.Google Scholar
  111. ES00a.
    W. E. Weinan and Ya. Sinai. New results on mathematical and statistical hydrodynamics. Russian Math. Surveys, 55(4):635–666, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  112. ES00b.
    W. E. Weinan and Ya. Sinai. Recent results on mathematical and statistical hydrodynamics. Russ. Math. Surv., 55(4):635–666, 2000.zbMATHCrossRefGoogle Scholar
  113. Fef06.
    C. L. Fefferman. Existence and smoothness of the Navier–Stokes equation. In The millennium prize problems, pages 57–67. Clay Math. Inst., Cambridge, MA, 2006.Google Scholar
  114. Fer97a.
    B. Ferrario. Ergodic results for stochastic Navier–Stokes equation. Stochastics Stochastics Rep., 60(3-4):271–288, 1997.zbMATHMathSciNetGoogle Scholar
  115. Fer97b.
    B. Ferrario. The Bénard problem with random perturbations: Dissipativity and invariant measures. NoDEA, Nonlinear Differ. Equ. Appl., 4(1):101–121, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  116. Fer99.
    B. Ferrario. Stochastic Navier–Stokes equations: analysis of the noise to have a unique invariant measure. Ann. Mat. Pura Appl. (4), 177:331–347, 1999.MathSciNetCrossRefGoogle Scholar
  117. Fer01.
    B. Ferrario. Pathwise regularity of nonlinear Itô equations: Application to a stochastic Navier–Stokes equation. Stochastic Anal. Appl., 19(1):135–150, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  118. Fer03.
    B. Ferrario. Uniqueness result for the 2D Navier–Stokes equation with additive noise. Stoch. Stoch. Rep., 75(6):435–442, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  119. Fer06.
    B. Ferrario. On some problems of regularity in two-dimensional stochastic hydrodynamics. G. Da Prato et al. (eds.), Stochastic partial differential equations and applications – VII. Papers of the 7th meeting, Levico, Terme, Italy, January 5–10, 2004. Boca Raton, FL: Chapman & Hall/CRC. Lecture Notes in Pure and Applied Mathematics 245, 97–103, 2006.Google Scholar
  120. FG04.
    F. Flandoli and M. Gubinelli. Random currents and probabilistic models of vortex filaments. Seminar on Stochastic Analysis, Random Fields and Applications IV, p. 129–139, Progr. Probab., 58, Birkhäuser, Basel, 2004.Google Scholar
  121. FG05.
    F. Flandoli and M. Gubinelli. Statistics of a vortex filament model. Electron. J. Probab. 10 (25):865–900, 2005 (electronic).MathSciNetGoogle Scholar
  122. FG95.
    F. Flandoli and D. Gatarek. Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields, 102(3):367–391, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  123. FG98.
    F. Flandoli and F. Gozzi. Kolmogorov equation associated to a stochastic Navier–Stokes equation. J. Funct. Anal., 160(1):312–336, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  124. FGGT05.
    F. Flandoli, M. Gubinelli, M. Giacquinta, and V. M. Tortorelli. Stochastic currents. Stoch. Proc. Appl., 115:1583–1601, 2005.zbMATHCrossRefGoogle Scholar
  125. Fla.
    F. Flandoli. An introduction to 3D stochastic fluid dynamics. These Proceedings.Google Scholar
  126. Fla94.
    F. Flandoli. Dissipativity and invariant measures for stochastic Navier–Stokes equations. NoDEA Nonlinear Differential Equations Appl., 1(4):403–423, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  127. Fla02.
    F. Flandoli. On a probabilistic description of small scale structures in 3D fluids. Annales Inst. Henri Poincaré, Probab. and Stat, 38: 207–228, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  128. Fla03.
    F. Flandoli. Some remarks on a statistical theory of turbulent flows. Davies, I. M. (ed.) et al., Probabilistic methods in fluids. Proceedings of the Swansea 2002 workshop, Wales, UK, April 14–19, 2002. Singapore: World Scientific. 144–160, 2003.Google Scholar
  129. FM95.
    F. Flandoli and B. Maslowski. Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Comm. Math. Phys., 172(1):119–141, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  130. FMB03.
    U. Frisch, T. Matsumoto, and J. Bec. Singularities of Euler flow? Not out of the blue! J. Stat. Phys., 113(5–6):761–781, 2003.MathSciNetGoogle Scholar
  131. FMRT01.
    C. Foias, O. P. Manley, R. Rosa, and R. Temam. Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, 83, 2001.Google Scholar
  132. FR83.
    J. Fröhlich and D. Ruelle. Statistical mechanics of vortices in an inviscid two-dimensional fluid. Comm. Math. Phys., 87(1):1–36, 1983.CrossRefGoogle Scholar
  133. FR01.
    F. Flandoli and M. Romito. Statistically stationary solutions to the 3-D Navier–Stokes equation do not show singularities. Electron. J. Probab., 6(5):15pp, 2001.MathSciNetGoogle Scholar
  134. FR02.
    F. Flandoli and M. Romito. Probabilistic analysis of singularities for the 3D Navier–Stokes equations. Mathematica Bohemica, 127(2): 211–218, 2002.zbMATHMathSciNetGoogle Scholar
  135. Fre97.
    M. Freidlin. Probabilistic approach to the small viscosity asymptotics for Navier–Stokes equations. Nonlinear Anal., Theory Methods Appl., 30(7):4069–4076, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  136. Fri95.
    U. Frisch. Turbulence. The legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, 1995.zbMATHGoogle Scholar
  137. FS76.
    J. Fröhlich and E. Seiler. The massive Thirring-Schwinger model (QED2): convergence of perturbation theory and particle structure. Helv. Phys. Acta, 49(6):889–924, 1976.MathSciNetGoogle Scholar
  138. FY92.
    H. Fujita Yashima. Équations de Navier–Stokes stochastiques non homogènes et applications. Scuola Normale Superiore, Pisa, 1992.Google Scholar
  139. Gal76.
    G. Gallavotti. Problèmes ergodiques de la mécanique classique. Enseignement du troisième cycle de la Physique en Suisse Romande. École Polytechnique Fédérale, Lausanne, 1976.Google Scholar
  140. Gal01.
    G. Gallavotti. Foundations of Fluid Dynamics. Springer, Berlin, 2001.zbMATHGoogle Scholar
  141. GGM80.
    K. Goodrich, K. Gustafson, and B. Misra. On converse to Koopman’s lemma. Phys. A, 102(2):379–388, 1980.MathSciNetCrossRefGoogle Scholar
  142. Gla77.
    H. M. Glaz. Two attempts at modeling two-dimensional turbulence. In Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), pages 135–155. Lecture Notes in Math., Vol. 615. Springer, Berlin, 1977.Google Scholar
  143. Gla81.
    H. M. Glaz. Statistical behavior and coherent structures in two-dimensional inviscid turbulence. SIAM J. Appl. Math., 41(3):459–479, 1981.zbMATHMathSciNetCrossRefGoogle Scholar
  144. Gli03.
    Yu. E. Gliklikh. Deterministic viscous hydrodynamics via stochastic analysis on groups of diffeomorphisms. Methods Funct. Anal. Topol., 9(2):146–153, 2003.zbMATHMathSciNetGoogle Scholar
  145. GSS02.
    F. Gozzi, S. S. Sritharan, and A. Świech. Viscosity solutions of dynamic-programming equations for the optimal control of the two-dimensional Navier–Stokes equations. Arch. Ration. Mech. Anal., 163(4):295–327, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  146. Hab91.
    Z. Haba. Ergodicity and invariant measures of some randomly perturbed classical fields. J. Math. Phys., 32(12):3463–3472, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  147. Hey90.
    J. G. Heywood. Open problems in the theory of the Navier–Stokes equations for viscous incompressible flow. The Navier–Stokes equations theory and numerical methods, Proc. Conf., Oberwolfach/FRG 1988, Lect. Notes Math. 1431, 1-22 (1990)., 1990.Google Scholar
  148. HKPS93.
    T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit. White noise, An infinite-dimensional calculus, volume 253 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.zbMATHGoogle Scholar
  149. HM06.
    M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2), 164(3):993–1032, 2006.zbMATHMathSciNetCrossRefGoogle Scholar
  150. Hop52.
    E. Hopf. Statistical hydrodynamics and functional calculus. J. Rat. Mech. Anal., 1:87–123, 1952.MathSciNetGoogle Scholar
  151. JLM85.
    G. Jona-Lasinio and P. K. Mitter. On the stochastic quantization of field theory. Commun. Math. Phys., 101:409–436, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  152. Kat67.
    T. Kato. On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rational Mech. Anal., 25:188–200, 1967.zbMATHMathSciNetCrossRefGoogle Scholar
  153. Kim02.
    J. U. Kim. On the stochastic Euler equations in a two-dimensional domain. SIAM J. Math. Anal., 33(5):1211–1227 (electronic), 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  154. KM80.
    R. H. Kraichnan and D. Montgomery. Two-dimensional turbulence. Rep. Progr. Phys., 43(5):547–619, 1980.MathSciNetCrossRefGoogle Scholar
  155. Kot95.
    P. Kotelenez. A stochastic Navier–Stokes equation for the vorticity of a two-dimensional fluid. Ann. Appl. Probab., 5(4):1126–1160, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  156. KR07.
    H. Kawabi and M. Röckner. Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach. J. Funct. Anal., 242(2):486–518, 2007.zbMATHMathSciNetCrossRefGoogle Scholar
  157. KS01.
    S. Kuksin and A. Shirikyan. Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom., 4(2):147–195, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  158. KT01.
    H. Koch and D. Tataru. Well-posedness for the Navier–Stokes equations. Adv. Math., 157(1):22–35, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  159. Kuk04.
    S. B. Kuksin. The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys., 115:469–492, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  160. Kuk06.
    S. B. Kuksin. Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions. European Mathematical society (EMS), 2006.Google Scholar
  161. Kuo75.
    H. H. Kuo. Gaussian measures in Banach spaces. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 463.zbMATHGoogle Scholar
  162. Lio96.
    P.-L. Lions. Mathematical Topics in Fluid Mechanics, volume 1, Incompressible Models. Science Publ., Oxford, 1996.zbMATHGoogle Scholar
  163. Lio98.
    P.-L. Lions. On Euler equations and statistical physics. Cattedra Galileiana. [Galileo Chair]. Scuola Normale Superiore, Classe di Scienze, Pisa, 1998.Google Scholar
  164. LJS97.
    Y. Le Jan and A. S. Sznitman. Stochastic cascades and 3-dimensional Navier–Stokes equations. Probab. Theory Relat. Fields, 109(3): 343–366, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  165. LM01.
    P.-L. Lions and N. Masmoudi. From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal., 158(3):173–193, 195–211, 2001.Google Scholar
  166. LNP00.
    J. A. León, D. Nualart, and R. Pettersson. The stochastic Burgers equation: finite moments and smoothness of the density. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 3(3):363–385, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  167. LOR94a.
    N. N. Leonenko, E. Orsingher, and K. V. Rybasov. Limiting distributions of the solutions of the multidimensional Burgers equation with random initial conditions. II. Ukr. Mat. Zh., 46(8):1003–1010, 1994.MathSciNetCrossRefGoogle Scholar
  168. LOR94b.
    N. N. Leonenko, E. Orsingher, and K. V. Rybasov. Limiting distributions of the solutions of the multidimensional Burgers equation with random initial conditions. I. Ukr. Math. J., 46(7):953–962, 1994.MathSciNetCrossRefGoogle Scholar
  169. LR98.
    V. Liskevich and M. Röckner. Strong uniqueness for certain infinite-dimensional Dirichlet operators and applications to stochastic quantization. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27(1):69–91, 1998.zbMATHMathSciNetGoogle Scholar
  170. LR02.
    P. G. Lemarié-Rieusset. Recent developments in the Navier–Stokes problem. Chapman & Hall Research Notes in Mathematics 431. Boca Raton, FL: Chapman & Hall/ 395 p., 2002.Google Scholar
  171. LW01.
    N. N. Leonenko and W. A. Woyczynski. Parameter identification for stochastic Burgers’ flows via parabolic rescaling. Probab. Math. Stat., 21(1):1–55, 2001.zbMATHMathSciNetGoogle Scholar
  172. Mas00.
    N. Masmoudi. Asymptotic problems and compressible-incompressible limit. Málek, Josef (ed.) et al., Advances in mathematical fluid mechanics. Lecture notes of the 6th international school on mathematical theory in fluid mechanics, Paseky, Czech Republic, September 19–26, 1999. Berlin: Springer. 119–158, 2000.Google Scholar
  173. Mat99.
    J. C. Mattingly. Ergodicity of 2D Navier–Stokes equations with random forcing and large viscosity. Commun. Math. Phys., 206(2): 273–288, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  174. MB02.
    A. J. Majda and A. L. Bertozzi. Vorticity and incompressible flow. Cambridge Univ. Press, 2002.Google Scholar
  175. MEF72.
    J. E. Marsden, D. Ebin, and A. Fischer. Diffeomorphism groups, hydrodynamics and relativity. Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress Differential Geometry and Applications, 1:135–279, 1972.MathSciNetGoogle Scholar
  176. Mel00.
    S. Meléard. A trajectorial proof of the vortex method for the two-dimensional Navier–Stokes equation. Ann. Appl. Probab., 10(4): 1197–1211, 2000.zbMATHMathSciNetGoogle Scholar
  177. Min00.
    R. A. Minlos. Introduction to mathematical statistical physics, volume 19 of University Lecture Series. American Mathematical Society, Providence, RI, 2000.zbMATHGoogle Scholar
  178. MP94.
    C. Marchioro and M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994.zbMATHGoogle Scholar
  179. MPP78.
    C. Marchioro, A. Pellegrinotti, and M. Pulvirenti. Selfadjointness of the Liouville operator for infinite classical systems. Comm. Math. Phys., 58(2):113–129, 1978.zbMATHMathSciNetCrossRefGoogle Scholar
  180. MR04.
    R. Mikulevicius and B. L. Rozovskii. Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal., 35(5):1250–1310 (electronic), 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  181. MR05.
    R. Mikulevicius and B. L. Rozovskii. Global L 2-solutions of stochastic Navier–Stokes equations. Ann. Probab., 33(1):137–176, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  182. MS02.
    J.-L. Menaldi and S. S. Sritharan. Stochastic 2-D Navier–Stokes equation. Appl. Math. Optimization, 46(1):31, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  183. MV94.
    H. P. McKean and K. L. Vaninsky. Statistical mechanics of nonlinear wave equations. In Trends and perspectives in applied mathematics, volume 100 of Appl. Math. Sci., pages 239–264. Springer, New York, 1994.Google Scholar
  184. MV00.
    R. Mikulevicius and G. Valiukevicius. On stochastic Euler equation in R d. Electron. J. Probab., 5(6) 20 pp. (electronic), 2000.Google Scholar
  185. NPe01.
    J. Neustupa, P. Penel, (eds.), Mathematical fluid mechanics. Recent results and open questions. Advances in Mathematical Fluid Mechanics. Basel: Birkhäuser. ix, 2001.Google Scholar
  186. NY03.
    B. Nachtergaele and H.-T. Yau. Derivation of the Euler equations from quantum dynamics. Comm. Math. Phys., 243(3):485–540, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  187. Osa87.
    H. Osada. Propagation of chaos for the two dimensional Navier–Stokes equation. Probabilistic methods in mathematical physics, Proc. Taniguchi Int. Symp., Katata and Kyoto/Jap. 1985, 303–334, 1987.MathSciNetGoogle Scholar
  188. Oss05.
    M. Ossiander. A probabilistic representation of solutions of the incompressible Navier–Stokes equations in 3. Probab. Theory Relat. Fields, 133(2):267–298, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  189. Pes85.
    C. S. Peskin. A random-walk interpretation of the incompressible Navier–Stokes equations. Commun. Pure Appl. Math., 38:845–852, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  190. Pul89.
    M. Pulvirenti. On invariant measures for the 2-D Euler flow. In: Mathematical aspects of vortex dynamics. Proceedings of the workshop held in Leesburg, Virginia, April 1988. R. Caflisch, (ed.), p. 88–96, SIAM, Philadelphia, PA, 1989.Google Scholar
  191. QY98.
    J. Quastel and H.-T. Yau. Lattice gases, large deviations, and the incompressible Navier–Stokes equations. Ann. of Math., 148(1): 51–108, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  192. Rap02a.
    D. L. Rapoport. On the geometry of the random representations for viscous fluids and a remarkable pure noise representation. Rep. Math. Phys., 50(2):211–250, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  193. Rap02b.
    D. L. Rapoport. Random diffeomorphisms and integration of the classical Navier–Stokes equations. Rep. Math. Phys., 49(1):1–27, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  194. Rap03.
    D. L. Rapoport. Random symplectic geometry and the realizations of the random representations of the Navier–Stokes equations by ordinary differential equations. Random Oper. Stoch. Equ., 11(4):371–401, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  195. Rap05.
    D. L. Rapoport. On the unification of geometric and random structures through torsion fields: Brownian motions, viscous and magneto-fluid dynamics. Found. Phys., 35:1205–1244, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  196. Rob91.
    R. Robert. A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys., 65(3-4):531–553, 1991.zbMATHCrossRefGoogle Scholar
  197. Rob03.
    R. Robert. Statistical hydrodynamics (Onsager revisited). Friedlander, S. (ed.) et al., Handbook of mathematical fluid dynamics. Vol. II. Amsterdam: North-Holland. 1–54, 2003.CrossRefGoogle Scholar
  198. Rom04.
    M. Romito. Ergodicity of the finite-dimensional approximation of the 3D Navier–Stokes equations forced by a degenerate noise. J. Stat. Phys., 114(1–2):155–177, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  199. Ros06.
    R. M. S. Rosa. Turbulence Theories. In: Encyclopedia of Mathematical Physics, J.-P. Françoise, G. L. Naber and S. T. Tsou (eds.), Elsevier, Oxford, Vol. 5, 295–302 p., 2006.Google Scholar
  200. Roz03.
    O. S. Rozanova. Solutions with linear profile of velocity to the Euler equations in several dimensions. Hou, Thomas Y. (ed.) et al., Hyperbolic problems: Theory, numerics, applications. Proceedings of the ninth international conference on hyperbolic problems, Pasadena, CA, USA, March 25–29, 2002. Berlin: Springer. 861–870, 2003.Google Scholar
  201. Roz04.
    O. S. Rozanova. Development of singularities for the compressible Euler equations with external force in several dimensions. Preprint at arXiv:math/0411652v2, December 2004.Google Scholar
  202. RS75.
    M. Reed and B. Simon. Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.Google Scholar
  203. RS91.
    R. Robert and J. Sommeria. Statistical equilibrium states for two-dimensional flows. J. Fluid Mech., 229:291–310, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  204. RS06.
    M. Röckner and Z. Sobol. Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. Ann. Probab., 34(2): 663–727, 2006.zbMATHMathSciNetCrossRefGoogle Scholar
  205. Rue69.
    D. Ruelle. Statistical Mechanics: Rigorous results. The Mathematical Physics Monographs Series. New York-Amsterdam: W. A. Benjamin, 1969.zbMATHGoogle Scholar
  206. Ser84a.
    D. Serre. Invariants et dégénérescence symplectique de l’équation d’Euler des fluides parfaits incompressibles. C. R. Acad. Sci. Paris Sér. I Math., 298(14):349–352, 1984.zbMATHMathSciNetGoogle Scholar
  207. Ser84b.
    D. Serre. Les invariants du premier ordre de l’équation d’Euler en dimension trois. Phys. D, 13(1-2):105–136, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  208. Shn97.
    A. Shnirelman. On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math., 50(12):1261–1286, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  209. Sim74.
    B. Simon. The P(φ) 2 Euclidean (quantum) field theory. Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics.Google Scholar
  210. Sin91.
    Ya. G. Sinai. Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Statist. Phys. 64 (1–2):1–12, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  211. Sin05a.
    Ya. Sinai. On local and global existence and uniqueness of solutions of the 3D Navier–Stokes system on 3. In Perspectives in analysis, volume 27 of Math. Phys. Stud., pages 269–281. Springer, Berlin, 2005.CrossRefGoogle Scholar
  212. Sin05b.
    Ya. Sinai. Power series for solutions of the 3D-Navier–Stokes system on R 3. J. Stat. Phys., 121(5–6):779–803, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  213. Sin08.
    Ya. Sinai, These proceedings.Google Scholar
  214. Soh01.
    H. Sohr. The Navier–Stokes equations. An elementary functional analytic approach. Birkhäuser Advanced Texts. Basel: Birkhäuser. 2001.Google Scholar
  215. Sta99.
    W. Stannat. (Nonsymmetric) Dirichlet operators on L 1: existence, uniqueness and associated Markov processes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28(1):99–140, 1999.Google Scholar
  216. Sta03.
    W. Stannat. L 1-uniqueness of regularized 2D-Euler and stochastic Navier–Stokes equations. J. Funct. Anal., 200(1):101–117, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  217. Sta07.
    W. Stannat. A new a priori estimate for the Kolmogorov operator of a 2D-stochastic Navier-Stokes equation. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 10(4): 483–497, 2007.zbMATHMathSciNetCrossRefGoogle Scholar
  218. Swa71.
    H.S.G. Swann. The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R 3. Trans. Am. Math. Soc., 157:373–397, 1971.zbMATHMathSciNetCrossRefGoogle Scholar
  219. SZ89.
    S. F. Shandarin and Ya. B. Zel′dovich. The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Modern Phys., 61(2):185–220, 1989.MathSciNetCrossRefGoogle Scholar
  220. Szn87.
    A. S. Sznitman. A propagation of chaos result of Burgers’ equation. Hydrodynamic behavior and interacting particle systems, Proc. Workshop, Minneapolis/Minn. 1986, IMA Vol. Math. Appl. 9, 181–188, 1987.Google Scholar
  221. Tem83.
    R. Temam. Navier–Stokes equations and nonlinear functional analysis, volume 41 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983.Google Scholar
  222. Tem84.
    R. Temam. Navier–Stokes equations, volume 2 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, third edition, 1984. Theory and numerical analysis, with an appendix by F. Thomasset.Google Scholar
  223. Tem00.
    R. Temam. Some developments on Navier–Stokes equations in the Second Half of the 20th Century. Development of mathematics. Université Paris-Sud, Orsay, 2000.Google Scholar
  224. TRW03.
    A. Truman, C.N. Reynolds, and D. Williams. Stochastic Burgers equation in d-dimensions – a one-dimensional analysis: hot and cool caustics and intermittence of stochastic turbulence. Davies, I. M. (ed.) et al., Probabilistic methods in fluids. Proceedings of the Swansea 2002 workshop, Wales, UK, April 14–19, 2002. Singapore: World Scientific. 239–262, 2003.Google Scholar
  225. TW03.
    A. Truman and J.-L. Wu. Stochastic Burgers equation with Lévy space-time white noise. Davies, I. M. (ed.) et al., Probabilistic methods in fluids. Proceedings of the Swansea 2002 workshop, Wales, UK, April 14–19, 2002. Singapore: World Scientific. 298–323, 2003.Google Scholar
  226. TW06.
    A. Truman and J.-L. Wu. Fractal Burgers’ equation driven by Lévy noise. Stochastic partial differential equations and applications, 7:295–310, 2006.MathSciNetGoogle Scholar
  227. TZ03.
    A. Truman and H. Z. Zhao. Burgers equation and the WKB-Langer asymptotic L 2 approximation of eigenfunctions and their derivatives. Davies, I. M. (ed.) et al., Probabilistic methods in fluids. Proceedings of the Swansea 2002 workshop, Wales, UK, April 14–19, 2002. Singapore: World Scientific. 332–366, 2003.Google Scholar
  228. VF88.
    M. J. Vishik and A. V. Fursikov. Mathematical problems of statistical hydromechanics. Mathematics and Its Applications: Soviet Series, 9, 576 p. Kluwer Academic Publishers, Dordrecht, Boston, London, 1988. Transl. from the Russian by D. A. Leites.Google Scholar
  229. VKF79.
    M. I. Vishik, A. I. Komech, and A.V. Fursikov. Some mathematical problems of statistical hydromechanics. Russ. Math. Surv., 34: 149–234, 1979.zbMATHMathSciNetCrossRefGoogle Scholar
  230. Woy98.
    W. A. Woyczyński. Burgers-KPZ turbulence. Göttingen Lectures. Lecture Notes in Mathematics. 1700. Berlin: Springer. xi, 318 p., 1998.Google Scholar
  231. Zgl03.
    P. Zgliczyński. Trapping regions and an ODE-type proof of the existence and uniqueness theorem for Navier–Stokes equations with periodic boundary conditions on the plane. Univ. Iagel. Acta Math, 41:89–113, 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sergio Albeverio
    • 1
  • Benedetta Ferrario
    • 2
  1. 1.Institut för Angewandte MathematikUniversität BonnGermany
  2. 2.Dipartimento di Matematica “F. Casorati”Università di PaviaItaly

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